Properties

Degree 2
Conductor $ 2^{2} \cdot 5^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·7-s + 9-s − 11-s + 4·13-s − 4·19-s + 8·21-s + 6·23-s − 4·27-s − 6·29-s + 8·31-s − 2·33-s − 2·37-s + 8·39-s + 6·41-s − 8·43-s − 6·47-s + 9·49-s + 6·53-s − 8·57-s − 12·59-s + 2·61-s + 4·63-s + 10·67-s + 12·69-s − 12·71-s + 16·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.917·19-s + 1.74·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s − 0.328·37-s + 1.28·39-s + 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 1.22·67-s + 1.44·69-s − 1.42·71-s + 1.87·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{1100} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1100,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.791104646\)
\(L(\frac12)\)  \(\approx\)  \(2.791104646\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.22980226171420, −18.74401741000899, −17.97726838205487, −17.30288913353413, −16.57968983706566, −15.43988577506529, −15.01468066683801, −14.48862202641743, −13.61941841829331, −13.29097731207038, −12.18179629683615, −11.10559329604952, −10.91830567582432, −9.652979243971770, −8.751180023146232, −8.300774525279624, −7.739813333822241, −6.624408986279670, −5.455873387559552, −4.530250898159293, −3.554172696492878, −2.455030795642607, −1.440856579824848, 1.440856579824848, 2.455030795642607, 3.554172696492878, 4.530250898159293, 5.455873387559552, 6.624408986279670, 7.739813333822241, 8.300774525279624, 8.751180023146232, 9.652979243971770, 10.91830567582432, 11.10559329604952, 12.18179629683615, 13.29097731207038, 13.61941841829331, 14.48862202641743, 15.01468066683801, 15.43988577506529, 16.57968983706566, 17.30288913353413, 17.97726838205487, 18.74401741000899, 19.22980226171420

Graph of the $Z$-function along the critical line